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<p>之前提到<em>Auto-regression</em>的decoding方法使得<em>transformer</em>在推理上的表现很慢，所以很多研究者在这方面做了很多研究，本文就介绍一个使用<em>Non-Auto Regression</em>的方法——<strong>Discrete Latent Variable</strong>。该方法与<em>Auto-regression</em>方法相比，效果上要稍差 一些，但是取得了比其他<em>Non-auto regression</em>方法都好的结果，而效率上也有很大的提升。</p>
<a id="more"></a>
<h1 id="1-简介"><a href="#1-简介" class="headerlink" title="1. 简介"></a>1. 简介</h1><h2 id="1-1-Auto-Regression"><a href="#1-1-Auto-Regression" class="headerlink" title="1.1 Auto-Regression"></a>1.1 Auto-Regression</h2><p><em>RNN</em>在机器翻译领域有着非常重要的应用，但是它本身由于不能进行并行计算，限制了它的效率，所以后来有些研究者希望能用<em>CNN</em>替代<em>RNN</em>。而<em>Transformer</em>的横空出世，使得机器翻译在训练效果和效率上都上了一个台阶，但是仍然存在一个问题。</p>
<p><em>Transformer</em>在生成一个序列的时候，通常需要根据之前的序列来预测下一个词，即当预测$y_n$时，需要利用$y_1, y_2, …, y_{n-1}$作为模型的输入。所以<em>transformer</em>在生成序列的时候是一个词一个词的生成，每生成一个词就需要进行一次推理，因此造成效率很低。这也就是所谓的<em>Auto-regression</em>问题。而<em>transformer</em>的<em>auto-regression</em>问题比<em>RNN</em>和<em>CNN</em>更加严重，因为<em>RNN</em>是根据前一个状态预测下一个状态，<em>CNN</em>是根据前<em>K</em>（kernel大小）个状态预测下一个状态，而<em>transformer</em>则是利用之前的所有状态预测下一个状态。虽然<em>transformer</em>在训练的时候可以很高效的训练，这是因为训练时的输出序列都已知，所以不需要<em>auto-regression</em>；但在进行decoding的时候输出是未知的，必须进行<em>auto-regression</em>，所以效率反而更低。</p>
<h2 id="1-2-Latent-Transformer"><a href="#1-2-Latent-Transformer" class="headerlink" title="1.2 Latent Transformer"></a>1.2 Latent Transformer</h2><p>为了克服<em>Auto-regression</em>问题，<a href="https://arxiv.org/pdf/1803.03382.pdf" target="_blank" rel="noopener">Kaiser et al. 2018</a>提出使用离散隐变量方法加速decoding推理。这种方法算不上真正解决了<em>Auto-regression</em>问题，但是算是对问题进行了优化吧，或者应该叫做<em>Semi-auto regression</em>。</p>
<p>这种方法简而言之就是，先用<em>auto-regression</em>生成一个固定长度的短的序列$l= \{l_1, l_2, …, l_m\}$，其中$m&lt;n$，然后再用$l$并行生成$y = \{y_1, y_2, …, y_n\}$。为了实现这种方法，我们需要变分自编码器。由于句子序列是离散的序列，在使用离散隐变量的时候会遇到不可求导的问题，因此如何解决这个问题就需要一些离散化的技术了。</p>
<h1 id="2-离散化技术"><a href="#2-离散化技术" class="headerlink" title="2. 离散化技术"></a>2. 离散化技术</h1><p>我们主要介绍四种离散化技术：</p>
<ul>
<li>Gumbel-softmax (<a href="http://arxiv.org/abs/1611.01144" target="_blank" rel="noopener">Jang et al., 2016;</a> <a href="https://arxiv.org/abs/1611.00712" target="_blank" rel="noopener">Maddison et al., 2016</a>)</li>
<li>Improved Semantic Hashing (<a href="https://arxiv.org/abs/1801.09797" target="_blank" rel="noopener">Kaiser &amp; Bengio, 2018</a>)</li>
<li>VQ-VAE (<a href="http://arxiv.org/abs/1711.00937" target="_blank" rel="noopener">van den Oord et al., 2017</a>)</li>
<li>Decomposed Vector Quantization</li>
</ul>
<p>给定目标序列$y=\{y_1, y_2, …, y_n\}$，将$y$输入到一个编码器（自编码器中的编码器，并非机器翻译模型中的编码器，下文的解码器同理，如非特殊说明<em>encoder</em>和<em>decoder</em>指的都是自编码器中的编码器和解码器）中产生一个隐变量表示$enc(y) \in \mathbb{R}^D$，其中$D$是隐变量空间的维度。令$K$为隐变量空间的大小，$[K]$表示集合$\{1, 2, …, K\}$。将连续隐变量$enc(y)$传入到一个<em>discretization bottleneck</em>中产生离散隐变量$z_d(y) \in [K]$，然后输入$z_q(y)$到解码器$dec$中。对于整数$i, m$我们使用$\tau_m(i)$代表用$m$ bits表示的二进制$i$，即用$\tau_m^{-1}$将$i$从二进制转换成 十进制。</p>
<p>下面我们主要介绍<em>discretization bottleneck</em>涉及到的离散化技术。</p>
<blockquote>
<p>实际上离散化技术是一个在VAE、GAN、RL中都有很重要应用的技术，本文只简单介绍它在文本生成方向的应用，而涉及到技术细节以及数学原理等更加详细的内容，以后会专门讨论，这里只说怎么用不说为什么。</p>
</blockquote>
<h2 id="2-1-Gumbel-Softmax"><a href="#2-1-Gumbel-Softmax" class="headerlink" title="2.1 Gumbel-Softmax"></a>2.1 Gumbel-Softmax</h2><p>将连续隐变量$enc(y)$变成离散隐变量的方法如下：</p>
<script type="math/tex; mode=display">
l = W enc(y) , W \in \mathbb{R}^{K\times D}</script><script type="math/tex; mode=display">
z_d(y) = \mathrm{arg} \max_{i\in[K]}~ l_i</script><ul>
<li>评估和推理时</li>
</ul>
<script type="math/tex; mode=display">
z_q(y) = e_j</script><p>其中$e \in \mathbb{R}^{K \times D}$，类似词向量的查询矩阵；$j=z_d(y)$。这一步相当于编码器生成一个短句子序列，然后这个短句子序列作为解码器的输入，通过查询词向量矩阵将句子中的词变成向量。</p>
<ul>
<li>训练时</li>
</ul>
<p>使用<em>Gumbel-softmax</em>采样生成$g_1, g_2, …, g_K$个独立同分布的<em>Gumbel</em>分布样本：</p>
<script type="math/tex; mode=display">
g_i \sim -\log(-\log(u))</script><p>其中$u \sim U(0,1)$表示均匀分布。然后用下式计算<em>softmax</em>得到$w \in \mathbb{R}^K$:</p>
<script type="math/tex; mode=display">
w_i = \frac{\exp((l_i+g_i)/\tau)}{\sum_i\exp((l_i+g_i)/\tau)}</script><p>得到$w$以后我们就可以简单地用：</p>
<script type="math/tex; mode=display">
z_q(y) = we</script><p>来获得$z_q(y)$。</p>
<p>注意<em>Gumbel-softmax</em>是可导的，也就是说我们可以直接通过后向传播对模型进行训练。</p>
<h2 id="2-2-Improved-Semantic-Hashing"><a href="#2-2-Improved-Semantic-Hashing" class="headerlink" title="2.2 Improved Semantic Hashing"></a>2.2 Improved Semantic Hashing</h2><p><em>Improved Semantic Hashing</em>主要来源于<a href="https://www.cs.utoronto.ca/~rsalakhu/papers/semantic_final.pdf" target="_blank" rel="noopener">Salakhutdinov &amp; Hinton, 2009</a>提出的<em>Semantic Hahsing</em>算法。</p>
<script type="math/tex; mode=display">
\sigma'(x) = \max(0, \min(1, 1.2\sigma(x)-0.1))</script><p>这个公式称为<em>饱和sigmoid</em>函数（<a href="https://arxiv.org/pdf/1511.08228.pdf" target="_blank" rel="noopener">Kaiser &amp; Sutskever, 2016;</a> <a href="https://papers.nips.cc/paper/6295-can-active-memory-replace-attention.pdf" target="_blank" rel="noopener">Kaiser &amp; Bengio, 2016</a>），</p>
<ul>
<li>训练时</li>
</ul>
<p>在$z_e(y) = enc(y)$中加入高斯噪声$\eta \sim \mathcal{N}(0,1)^D$，然后传入给饱和sigmoid函数</p>
<script type="math/tex; mode=display">
f_e(y) = \sigma'(z_e(y) + \eta)</script><p>使用下式将$f_e(y)$进行离散化：</p>
<p><img src="https://img.vim-cn.com/98/5790fa5e2da74037e22462e5dfb60e07035bd8.png" alt></p>
<p>解码器的输入用两个嵌入矩阵计算$e^1, e^2 \in \mathbb{R}^{K \times D}$：</p>
<script type="math/tex; mode=display">
z_q(y) = e^1_{h_{e(y)}}+e^2_{1-h_{e(y)}}</script><p>其中$h_{e}$是从$f_e$或者$g_e$中随机选择的。</p>
<ul>
<li>推理时</li>
</ul>
<p>令$f_e=g_e$</p>
<h2 id="2-3-Vector-Quantization"><a href="#2-3-Vector-Quantization" class="headerlink" title="2.3 Vector Quantization"></a>2.3 Vector Quantization</h2><p><em>Vector Quantized - Variational Autoencoder (VQ-VAE)</em>是<a href="http://arxiv.org/abs/1711.00937" target="_blank" rel="noopener">van denOord et al., 2017</a>提出的一种离散化方法。<em>VQ-VAE</em>的基本方法是使用最近邻查找矩阵$e \in \mathbb{R}^{K\times D}$将$enc(y)$进行数值量化。具体方法如下：</p>
<script type="math/tex; mode=display">
z_q = e_k, k=\mathrm{arg} \min_{j\in [K]} \|enc(y) -e_j \|_2</script><p>对应的离散化隐变量$z_d(y)$是$e$矩阵中与$enc(y)$距离$k$索引最近的值。损失函数定义如下：</p>
<script type="math/tex; mode=display">
L = l_r +\beta\|enc{y}-sg(z_q(y)) \|_2</script><p>其中$sg(\cdot)$定义如下：</p>
<p><img src="https://img.vim-cn.com/cd/12fa252851f7de219a8e5f3334759406d5a27c.png" alt></p>
<p>$l_r$即为给定$z_q(y)$后模型的损失（比如交叉熵损失等）。</p>
<p>使用下面两个步骤获得<em>exponential moving average (EMA)</em>：</p>
<ol>
<li>每个$j \in [K]$都用$e_j$；</li>
<li>统计编码器隐状态中使用$e_j$作为最近邻量化的个数$c_j$。</li>
</ol>
<p>$c_j$的更新方法如下：</p>
<script type="math/tex; mode=display">
c_j \leftarrow \lambda c_j+(1-\lambda)\sum_l 1[z_q(y_l)=e_j]</script><p>然后对$e_j$进行更新：</p>
<script type="math/tex; mode=display">
e_j \leftarrow \lambda e_j +(1+\lambda)\sum_l \frac{1[z_q(y_l)=e_j]enc(y_l)}{c_j}</script><p>其中$1[\cdot]$是一个指示函数，$\lambda$是延迟参数，实验中设置为$0.999$。</p>
<h2 id="2-4-Decomposed-Vector-Quantization"><a href="#2-4-Decomposed-Vector-Quantization" class="headerlink" title="2.4 Decomposed Vector Quantization"></a>2.4 Decomposed Vector Quantization</h2><p>当离散隐变量空间很大的时候<em>VQ-VAE</em>会有一个问题——<em>index collapse</em>：由于“富人越富，穷人越穷”效应，只有少数的嵌入向量能得到训练。</p>
<p>具体来说就是如果一个嵌入向量$e_j$距离很多编码器的输出$enc(y_1), enc(y_2), …, enc(y_i)$都很近，那么它就能通过上面$c_j$和$e_j$的更新更加靠近，到最后只有少数几个嵌入向量被用到。因此，本文提出了一个<em>VQ-VAE</em>的变种——<em>DVQ</em>使$K$值很大的时候也能做到充分利用嵌入向量。</p>
<h3 id="2-4-1-Sliced-Vector-Quantization"><a href="#2-4-1-Sliced-Vector-Quantization" class="headerlink" title="2.4.1 Sliced Vector Quantization"></a>2.4.1 Sliced Vector Quantization</h3><p><em>Sliced vector quantization</em>顾名思义，就是将$enc(y)$切成$n_d$个小的切片：</p>
<script type="math/tex; mode=display">
enc^1(y)\odot enc^2(y)...\odot enc^{n_d}(y)</script><p>其中每一个$enc(y)$的维度为$D/N_d$，$\odot$表示拼接。</p>
<h3 id="2-4-2-Projected-Vector-Quantization"><a href="#2-4-2-Projected-Vector-Quantization" class="headerlink" title="2.4.2 Projected Vector Quantization"></a>2.4.2 Projected Vector Quantization</h3><p>另一个方法是，使用固定的随机初始化投影集合：</p>
<script type="math/tex; mode=display">
\{ \pi^i \in \mathbb{R}^{D\times D/n_d} | i \in [n_d]\}</script><p>将$enc(y)$投影到$R^{D/n_d}$的向量空间中去。</p>
<h1 id="3-Latent-Transformer"><a href="#3-Latent-Transformer" class="headerlink" title="3. Latent Transformer"></a>3. Latent Transformer</h1><p>介绍了这么多离散化的技术，下面就需要将这些离散化的技术应用到模型中去。给定输入输出序列对：$(x, y) = (x_1, x_2, …, x_k, y_1, y_2, …, y_n)$，<em>Latent Transformer</em>包含下面三个部分：</p>
<ul>
<li>$ae(y, x)$函数用来对$y$进行编码成$l=l_1, l_2, …, l_m$；</li>
<li>使用<em>Transformer</em> （即$lp(x)$）对$l$进行预测</li>
<li>$ad(l, x)$函数并行化产生$y$</li>
</ul>
<p>损失函数分成两部分：</p>
<ol>
<li>$l_r = compare(ad(ae(y,x), x), y)$;</li>
<li>$l = compare(ae(y, x), lp(x))$</li>
</ol>
<script type="math/tex; mode=display">
L = l_r + l</script><h3 id="3-1-ae-y-x-函数"><a href="#3-1-ae-y-x-函数" class="headerlink" title="3.1 $ae(y,x)$函数"></a>3.1 $ae(y,x)$函数</h3><p><img src="https://cdn.jsdelivr.net/gh/rogerspy/blog-imgs/a8426346afc84ad83df607d7bfecdbd1be2f6b.png" alt></p>
<p>结构如图。其中<em>bottleneck</em>即为上面介绍的各种离散隐变量的方法。</p>
<h3 id="3-2-ad-y-x-函数"><a href="#3-2-ad-y-x-函数" class="headerlink" title="3.2 $ad(y, x)$函数"></a>3.2 $ad(y, x)$函数</h3><p><img src="https://cdn.jsdelivr.net/gh/rogerspy/blog-imgs/a21145d923a21d55c75575909194dd3d355f1f.png" alt></p>
<p>结构如图。</p>
<h1 id="4-实验结果"><a href="#4-实验结果" class="headerlink" title="4. 实验结果"></a>4. 实验结果</h1><p><img src="https://cdn.jsdelivr.net/gh/rogerspy/blog-imgs/8dcc671f32187d72fd9f063da5bbeeb5eee9ee.png" alt></p>
<p>图中作为<em>baseline</em>的<em>NAT</em>是<a href="http://arxiv.org/abs/1711.02281" target="_blank" rel="noopener">Gu et al. 2017</a>另一种<em>Non-auto regression</em>的方法。</p>
<h1 id="5-参考资料"><a href="#5-参考资料" class="headerlink" title="5. 参考资料"></a>5. 参考资料</h1><ol>
<li><p><a href="https://arxiv.org/pdf/1803.03382.pdf" target="_blank" rel="noopener">Fast Decoding in Sequence Models Using Discrete Latent Variables</a> Kaiser et al., 2018</p>
</li>
<li><p><a href="http://arxiv.org/abs/1611.01144" target="_blank" rel="noopener">Categorical reparameterization with gumbel-softmax</a> Jang et al. 2016</p>
</li>
<li><p><a href="http://arxiv.org/abs/1611.00712" target="_blank" rel="noopener">The concrete distribution: A continuous relaxation of discrete random variables</a> Maddison et al., 2016</p>
</li>
<li><p><a href="https://arxiv.org/abs/1610.08613" target="_blank" rel="noopener">Can active memory replace attention?</a> Kaiser, Łukasz and Bengio, Samy. 2016</p>
</li>
<li><a href="http://arxiv.org/abs/1801.09797" target="_blank" rel="noopener">Discrete autoencoders for sequence models</a> Kaiser, Łukasz and Bengio, Samy. 2018</li>
<li><a href="https://arxiv.org/abs/1511.08228" target="_blank" rel="noopener">Neural GPUs learn algorithms</a> Kaiser, Łukasz and Sutskever, Ilya. 2016</li>
<li><a href="http://arxiv.org/abs/1711.02281" target="_blank" rel="noopener">Non-autoregressive neural machine translation</a> Gu et al., 2017</li>
<li><a href="http://arxiv.org/abs/1711.00937" target="_blank" rel="noopener">Neural discrete representation learning</a> van den Oord et al., 2017</li>
<li><a href="https://www.cs.utoronto.ca/~rsalakhu/papers/semantic_final.pdf" target="_blank" rel="noopener"> Semantic hashing</a> Salakhutdinov, Ruslan and Hinton, Geoffrey E. 2009</li>
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      <ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#1-简介"><span class="toc-text">1. 简介</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-1-Auto-Regression"><span class="toc-text">1.1 Auto-Regression</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#1-2-Latent-Transformer"><span class="toc-text">1.2 Latent Transformer</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#2-离散化技术"><span class="toc-text">2. 离散化技术</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#2-1-Gumbel-Softmax"><span class="toc-text">2.1 Gumbel-Softmax</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-2-Improved-Semantic-Hashing"><span class="toc-text">2.2 Improved Semantic Hashing</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-3-Vector-Quantization"><span class="toc-text">2.3 Vector Quantization</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-4-Decomposed-Vector-Quantization"><span class="toc-text">2.4 Decomposed Vector Quantization</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#2-4-1-Sliced-Vector-Quantization"><span class="toc-text">2.4.1 Sliced Vector Quantization</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-4-2-Projected-Vector-Quantization"><span class="toc-text">2.4.2 Projected Vector Quantization</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#3-Latent-Transformer"><span class="toc-text">3. Latent Transformer</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#3-1-ae-y-x-函数"><span class="toc-text">3.1 $ae(y,x)$函数</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-2-ad-y-x-函数"><span class="toc-text">3.2 $ad(y, x)$函数</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#4-实验结果"><span class="toc-text">4. 实验结果</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#5-参考资料"><span class="toc-text">5. 参考资料</span></a></li></ol>
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